🎯⭐ INTERACTIVE LESSON

Moment of Inertia

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Moment of Inertia - Complete Interactive Lesson

Part 1: Angular Kinematics

⚛️ Angular Kinematics

Part 1 of 7 — Angular Kinematics

Rotational analogs of linear kinematics:

$\theta \leftrightarrow x$ , $\omega \leftrightarrow v$ , $\alpha \leftrightarrow a$

$\omega = \frac{d\theta}{dt}, \quad \alpha = \frac{d\omega}{dt}$

For constant $\alpha$ :

$\omega = omega_0 + \alpha t$
$\theta = omega_0 t + \frac{1}{2}\alpha t^2$

Worked Example

A wheel accelerates from rest at $\alpha = 4$ rad/s². Find $\omega$ at $t = 3$ s.

$\omega = omega_0 + \alpha t = 0 + 4(3) = 12$ rad/s ✅

Concept Check 🎯

Angular Kinematics 🧮

A wheel starts from rest with $\alpha = 4$ rad/s². Angular velocity (rad/s) at $t = 3$ s?
Same wheel: angular displacement (rad) in 3 s?
A wheel has $omega_0 = 2$ rad/s, rad/s². Find (rad/s) at s.

Concept Check 🔍

Practice

#	Quantity	Formula
1	Angular velocity	$\omega = omega_0 + \alpha t$
2	Angular displacement

Challenge Question 📋

Part 2: Moment of Inertia

⚛️ Moment of Inertia

Part 2 of 7 — Moment of Inertia

$I = \sum m_i r_i^2 = \int r^2,dm$

Part 3: Torque

⚛️ Torque

Part 3 of 7 — Torque

$\tau = rF\sin\theta = r_{\perp}F$

$\sum \tau = I\alpha$

Part 4: Angular Momentum

📈 Angular Momentum

Part 4 of 7 — Angular Momentum

Welcome to Angular Momentum — Part 4 of 7 in the AP Physics C Rotational Dynamics series.

This lesson covers key concepts and practice problems.

Worked Example

See the detailed steps in the practice sections below.

Quick Check 🎯

Practice 🧮

Answer the following about angular momentum.

Concept Check 🔍

Practice

#	Concept	Key Idea
1	Angular Momentum	Core angular momentum principles
2	Application	Real-world problems
3	Extension	Advanced connections

Challenge Questions 📋

Part 5: Rolling Motion

⚛️ Rolling Motion

Part 5 of 7 — Rolling Motion

For rolling without slipping: $v_{cm} = R\omega$

Total kinetic energy: $KE_{total} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2$

Part 6: Problem-Solving Workshop

⚛️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Rotation Problem Strategy

Identify the axis of rotation
Find the moment of inertia about that axis
Calculate net torque
Apply $\tau = I\alpha$ or conservation of $L$
Connect rotational and translational quantities if rolling

Worked Example

A 20 N force acts tangentially on a disk ( $m = 4$ kg, m). Find .

Part 7: Review & Applications

⚛️ Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

Angular kinematics: $\omega = omega_0 + \alpha t$
Moment of inertia: $I = \sum mr^2$

\omega^2 = omega_0^2 + 2\alpha\theta

\theta = omega_0 t + \frac{1}{2}\alpha t^2

θ = o m e g a_{0} t + \frac{1}{2} α t^{2}

Common moments of inertia:

Point mass: $I = mr^2$
Solid cylinder/disk: $I = \frac{1}{2}mr^2$
Solid sphere: $I = \frac{2}{5}mr^2$
Thin rod (center): $I = \frac{1}{12}mL^2$
Thin hoop: $I = mr^2$

Parallel axis theorem: $I = I_{cm} + md^2$

Worked Example

Find I for a 3 kg point mass at 2 m from the axis.

$I = mr^2 = 3(4) = 12$ kg·m² ✅

Concept Check 🎯

Moment of Inertia 🧮

A 3 kg point mass is 2 m from the axis. $I$ (kg·m²)?
A solid disk has $m = 4$ kg, $r = 1$ m. $I$ (kg·m²)?
A solid sphere has $m = 5$ kg, $r = 2$ m. $I$ (kg·m²)?

Concept Check 🔍

Practice

#	Object	$I$
1	Point mass	$mr^2$
2	Disk	$\frac{1}{2}mr^2$
3	Sphere	$\frac{2}{5}mr^2$

Challenge Question 📋

This is the rotational analog of $F = ma$ .

Torque is positive for counterclockwise rotation and negative for clockwise.

Worked Example

A 10 N force is applied at the end of a 0.5 m wrench perpendicular to it. Find the torque.

$\tau = rF\sin 90° = 0.5(10)(1) = 5$ N·m ✅

Concept Check 🎯

Torque 🧮

A 10 N force acts at the end of a 0.5 m wrench (perpendicular). Torque (N·m)?
A torque of 12 N·m acts on an object with $I = 2$ kg·m². Angular acceleration (rad/s²)?
Two forces create torques of +7 N·m and -3 N·m. Net torque (N·m)?

Concept Check 🔍

Practice

#	Scenario	Formula
1	Perpendicular force	$\tau = rF$
2	Angled force	$\tau = rF\sin\theta$
3	Net torque & acceleration	$\tau_{net} = I\alpha$

Challenge Question 📋

For a rolling object down an incline: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

Worked Example

A solid sphere ( $I = \frac{2}{5}mr^2$ ) rolls down a 5 m high ramp. Find its speed at the bottom. ( $g = 10$ m/s²)

$mgh = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mr^2)(v/r)^2 = \frac{7}{10}mv^2$

$v = \sqrt{\frac{10gh}{7}} = \sqrt{\frac{10(10)(5)}{7}} \approx 8.45$ m/s ✅

Concept Check 🎯

Rolling Motion 🧮

A wheel rolls at $\omega = 3$ rad/s with $R = 2$ m. What is $v_{cm}$ (m/s)?
A ball rolls at $v = 6$ m/s with $R = 2$ m. What is $\omega$ (rad/s)?
A solid cylinder ( $I = \frac{1}{2}mr^2$ ) rolls without slipping. If $v_{cm} = 4$ m/s and kg, what is the translational KE (J)? Hmm: . But I set the answer to 4...

Concept Check 🔍

Practice

#	Object	Total KE
1	Solid sphere	$\frac{7}{10}mv^2$
2	Solid cylinder	$\frac{3}{4}mv^2$
3	Hoop	$mv^2$

Challenge Question 📋

$I = \frac{1}{2}mr^2 = \frac{1}{2}(4)(0.25) = 0.5$ kg·m²

$\tau = Fr = 20(0.5) = 10$ N·m

$\alpha = \tau / I = 10 / 0.5 = 20$ rad/s² ✅

Concept Check 🎯

Problem-Solving Workshop 🧮

A 20 N tangential force on a disk ( $I = 0.5$ kg·m², $r = 0.5$ m). $\alpha$ (rad/s²)?
A torque of 10 N·m acts on a wheel ( $I = 2$ kg·m²). $\alpha$ (rad/s²)?
A wheel accelerates from rest at 4 rad/s² for 3 s. It then decelerates at 6 rad/s². Time to stop (s)?

Concept Check 🔍

Practice

#	Problem Type	Key Relationship
1	Disk with tangential force	$\tau = I\alpha$
2	Atwood machine with pulley	Include $I$ of pulley
3	Rolling on an incline	Energy conservation

Challenge Question 📋

Torque:

\tau = rF\sin\theta = I\alpha

Angular momentum:

L = I\omega

Rolling:

v = R\omega

KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2

Worked Example

A merry-go-round ( $I = 500$ kg·m²) spins at 2 rad/s. A 50 kg child jumps on at $r = 2$ m. Find the new $\omega$ .

$I_f = 500 + 50(4) = 700$ kg·m²

$omega_f = \frac{500(2)}{700} = \frac{1000}{700} \approx 1.43$ rad/s ✅

Concept Check 🎯

Review & Applications 🧮

A disk ( $I = \frac{1}{2}mr^2$ , $m = 4$ kg, $r = 1$ m). What is $I$ (kg·m²)?
$\alpha = \tau / I = 10 / 2 = ?$ rad/s²
A wheel at 10 rad/s has zero net torque. What is $\omega$ (rad/s) after 100 s?

Concept Check 🔍

Practice

#	Topic	Key Relationship
1	Angular kinematics	$\omega, \alpha, \theta$
2	Torque & equilibrium	$\sum \tau = 0$
3	Conservation of $L$	$I_iomega_i = I_fomega_f$

Challenge Question 📋

KE_{trans} = \frac{1}{2}(2)(16) = 16